Delft University of Technology

Delft Center for Systems and Control

Technical report 14-011

An analytic model for a 13C isotope

separation process by cryogenic

distillation∗

D.C. Dumitrache, I. Inoan, and B. De Schutter

If you want to cite this report, please use the following reference instead:

D.C. Dumitrache, I. Inoan, and B. De Schutter, “An analytic model for a 13C isotope

separation process by cryogenic distillation,” Journal of Process Control, vol. 24, no.

5, pp. 463–474, May 2014.

Delft Center for Systems and Control

Delft University of Technology

Mekelweg 2, 2628 CD Delft

The Netherlands

phone: +31-15-278.51.19 (secretary)

fax: +31-15-278.66.79

URL: http://www.dcsc.tudelft.nl

∗This report can also be downloaded via http://pub.deschutter.info/abs/14_011.html

An analytic model for a 13C isotope separation process by1

cryogenic distillation2

Dan Calin Dumitrachea,∗, Iulia Inoana, Bart De Schutterb3

aDepartment of Automation, Technical University of Cluj-Napoca, Memorandumului 28, 400114,4

Cluj-Napoca, Romania5

bDelft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft,6

The Netherlands7

Abstract8

In this paper we present a structured approach to the modeling of an isotope separation plant

that makes use of distillation principles for 13C isotope separation. In the first part of the paper,

after a brief review of isotope separation processes with an accent on isotope separation by

distillation, we define our initial-boundary-value problem, which is a partial differential equation.

By applying the Laplace transform to the partial differential equation that governs the evolution

of the desired isotope with respect to height and time, we obtain a linear homogeneous ordinary

differential equation. After solving the obtained ordinary differential equation, we use Heaviside’s

expansion theorem to find the inverse Laplace transform, and thus, the analytic model of the

isotope separation process follows. Using the analytic model it is possible to determine the

concentration of the desired isotope at any height, at any moment of time, with respect to plant

parameters and thus, expensive experiments can be avoided. In the second part of the paper we

use the analytic model to simulate the isotope separation process, followed by the assessment of

the results against experimental data.

Keywords: Mathematical modeling, Isotope separation, Cryogenic distillation, Analytic model,9

Simulation10

1. Introduction11

Isotope-based applications are met in a variety of fields such as chemistry, medicine, hydrol-12

ogy, and energy. The use of isotopic labeled compounds in order to explore different chemical,13

biological, or hydrological mechanisms is well known [1–4]. Likewise, the practical applications14

of radioisotopes in energy generation has been studied extensively, at this moment resulting in15

436 working nuclear plants worldwide and another 63 plants under construction [5–8]. Moreover,16

there is an increasing interest in the use of isotopes in medical investigations [9, 10]. Particularly,17

∗Corresponding author. Tel.: +40 740 365 167; Fax: +40 266 313 058;E-mail address: dan.dumitrache@aut.utcluj.roPreprint submitted to Elsevier

the 13C isotope plays an important role in oceanic and atmospheric studies, as well as in medical18

diagnosis based on breath CO2 tests (13C/12C ratio) used to avoid common invasive procedures19

[11, 12].20

In order to increase the concentration of a certain isotope, one can adopt an isotope separation21

method. Isotope separation techniques depend on several factors, among which are the properties22

of the chemical compound involved, the various applications that require different concentrations,23

and the cost of the isotope separation process [13–17].24

Urey [18] was among the first to show that the isotopic compounds do not only differ in ther-25

modynamical properties but also in their physical and chemical properties. In [19] Bigeleisen26

and Mayer proposed a calculation method for the equilibrium constant of the isotopic exchange27

reactions. Bigeleisen continued his work on isotopes providing a comprehensive review in [13]28

and a survey with emphasis on isotope effects in [15]. A description of various isotope separation29

techniques can be found in [4, 20–22]. Cohen has developed the general theory of multistage iso-30

tope separation processes in [23], also treating briefly the behavior of liquid-gas countercurrent31

chemical exchange towers. In [24] London provides an insightful study on isotope separation.32

Referring to 13C isotope separation processes, McInteer presents in [25] the design issues of a33

13C cryogenic rectification plant, while Andreev et al. [26] present the characteristics of different34

13C isotope separation plants. In [27], following first-principles knowledge, we have presented a35

structured modeling approach for a 13C isotope separation process that uses distillation princi-36

ples. We acknowledge the work of Zemansky [28] regarding thermal phenomena, while in the37

field of mass transfer and separation processes Cussler, King, and Benitez [29–31] are insightful38

references. Concerning the dynamics and control of distillation columns, Skogestad et al. [32–36]39

provide a detailed and comprehensible treatment. Regarding distributed parameter systems and40

partial differential equations we mention [37–41]. In the field of complex analysis and its appli-41

cations we acknowledge the work of Townsend, LePage, Duffy, and Brown [42–45], while in the42

field of numerical analysis [46–48] represent standard works.43

Isotope separation processes have, like most physical, chemical, or biological processes, a44

coupled time-space nature, and thus they belong to the class of distributed parameter systems45

[37, 38, 49, 50]. Modeling of a distributed parameter system, in general, and modeling of an46

isotope separation process in particular, is a time-consuming task and requires knowledge from47

different fields. Numerical methods are widely used in order to solve different mathematical48

models. However, closed-form solutions (i.e. analytical solutions) are preferred, since they are in49

general faster and allow a more meaningful analysis of the process and they ease the linking to50

2

the physics of the process [40, 51–54].51

The objective of this paper is to provide the analytic solution of the model of a 13C isotope52

separation process followed by the simulation of the isotope separation plant and hence, to provide53

a basis for future studies in optimization and process control design. The main contribution of54

this study consists in the comprehensive determination of the analytic model of the 13C isotope55

separation process by cryogenic distillation of carbon monoxide. This approach is applicable to56

other isotope separation processes as well. The assessment of the analytic model is made via57

simulation. Using the analytic model one can determine the concentration of desired isotope at58

any height and at any moment of time, with respect to plant parameters and thus, expensive59

experiments can be avoided. To the authors’ best knowledge we are the first to treat these issues60

for a 13C cryogenic distillation plant.61

The paper is structured as follows. In Section 2 we give a brief account of isotope separation62

processes with emphasis on isotope separation by distillation, followed by a short review of the63

experimental pilot-scale plant developed at the National Institute for Research and Development64

of Isotopic and Molecular Technologies in Cluj-Napoca, Romania. In Section 3, based on the65

linear partial differential equation that governs the evolution of the desired isotope concentration66

with respect to space and time, we derive the analytic model of the isotope separation process.67

In Section 4 we use the obtained analytic model to determine the evolution of the concentration68

of the desired isotopic compound at both ends of the distillation column. The concentration69

evolution is compared with experimental data. Next, we simulate the isotope concentration70

distribution in the column followed by the assessment of the results. Section 5 concludes the71

paper.72

The list of tables and figures present the tables and figures referred to within this paper. The73

most frequent notations used in this paper are listed in Table 1.74

List of Tables75

1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

2 Parameters of the analytical solution and the heights of the rectifying and stripping77

section in the case of all three isotope separation experiments. . . . . . . . . . . . 2378

3 The values of(

γ(sj)Zr

)

and(

γ(sj)Zs

)

when j = 2, 3, 4, 5, and 6 in the case of79

all three isotope separation experiments. . . . . . . . . . . . . . . . . . . . . . . . 2380

4 The values of the maximum and average relative error between simulated and81

measured data at the ends of the separation column. . . . . . . . . . . . . . . . . 2782

3

List of Figures83

1 Experimental pilot-scale cryogenic distillation plant with condenser K, distillation84

column C, reboiler B, vacuum jacket VJ, rough pump RP, diffusion pump DP,85

temperature sensors T1−T2, manometers M1−M3, level sensor L, feed reservoir86

FR, buffer tank BT, and waste reservoir WR. . . . . . . . . . . . . . . . . . . . . 787

2 Schematic representation of the 13C cryogenic distillation and of the column’s88

rectifying and stripping section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989

3 Graphical representation of the functions y1(

γ(s)Zr

)

, y1(

γ(s)Zs

)

and y2(

γ(s)Zr

)

,90

y2(

γ(s)Zs

)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1891

4 13C concentration evolution according to the analytical solution and 13C concen-92

trations achieved by the pilot-scale experimental plant. . . . . . . . . . . . . . . . 2493

5 Relative errors between simulated and measured data at the end of the rectifying94

section (bottom of the column) and at the end of the stripping section (top of the95

column). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2596

6 13C isotope concentration distribution with respect to time and height. . . . . . . 2697

2. Isotope separation process98

Isotope separation techniques have their origins in the isotope effects of different isotopic99

compounds that arise from the differences in the nuclear properties of the isotopes. Some prac-100

tical isotope separation methods are based on chemical exchange processes, gaseous diffusion,101

gas centrifuge, laser separation, electromagnetic isotope separation (which is a form of mass102

spectrometry), and distillation [21, 22].103

2.1. Isotope separation by cryogenic distillation104

In the case of light elements like boron, carbon, nitrogen, or oxygen, due to the large relative105

mass difference, distillation was successfully applied to the separation of isotopes [17, 24]. Separa-106

tion processes by distillation are based on the notion of relative volatility, which is a comparative107

measure for the vapor pressure of the components within a mixture [30, 32].108

Regarding the separation of carbon isotopes, it has been shown that the vapor pressure109

isotope effect is the highest in the case of carbon monoxide (CO) and methane (CH4). Thus,110

these substances are preferred as raw material in 13C isotope separation by distillation. Since111

both carbon monoxide and methane are gaseous in standard conditions, cryogenic distillation is112

applied [26, 55, 56].113

4

Table 1: Nomenclature

Latin symbols

F feed flow rate mol/(m2s)H hold-up per unit volume mol/m3

K volumetric overall mass transfer coefficient mol/(m3s)L liquid molar flow rate per unit area mol/(m2s)n 13CO mole fraction in vapor phase −

N 13CO mole fraction in liquid phase −

P product flow rate mol/(m2s)t time sV vapor molar flow rate per unit area mol/(m2s)W waste flow rate mol/(m2s)z height (position) m

Z total height m

Greek symbols

α relative volatility −

ε enrichment factor −

τ molar transfer rate per unit volume mol/(m3s)

Superscript

0 pure component −

13 atomic number −

Subscript

0 natural abundance −

13CO isotopic species −

c column −

F is referring to feed flow rate −

g general −

j index −

k index −

l liquid −

p particular −

P is referring to product flow rate −

r rectifying −

s stripping −

v vapor −

W is referring to waste flow rate −

5

Since this paper treats the modeling and simulation of a 13C separation plant by cryogenic114

distillation of carbon monoxide, in the following we will refer to carbon monoxide. Due to the115

fact that carbon has two stable isotopes, 12C and 13C with a natural abundance of 98.89%116

and 1.11% respectively, and oxygen has three stable isotope, 16O, 17O, and 18O with natural117

abundances of 99.76%, 0.04%, and 0.20% respectively, in the process of cryogenic distillation118

of carbon monoxide, six molecular species are involved. However, considering the low natural119

abundances of 17O and 18O, it is reasonable to relate the carbon monoxide to a binary mixture120

form of 12C16O and 13C16O [25]. The separation factor, when the 13CO (i.e. the less volatile121

component) is the product, is expressed by1 [32]:122

α =p012CO

p013CO

=N/(1−N)

n/(1− n)(1)

where p012CO and p013CO are the vapor pressure of the pure components 12CO and 13CO, N and123

n refer to the 13CO mole fraction in the liquid phase respectively vapor phase, while 1−N and124

1− n are the 12CO mole fraction in the liquid and vapor phase, respectively.125

An accepted value for α, at the boiling point of CO in normal conditions (i.e. 81.6K), is126

α = 1.0069 [26, 57].127

2.2. The pilot-scale experimental plant128

At the National Institute for Research and Development of Isotopic and Molecular Tech-129

nologies, Cluj-Napoca, Romania, an experimental 13C isotope separation plant has been de-130

veloped. The plant is shown in Fig. 1. The configuration consists of a single column of131

7000mm in height with an inner diameter of 16mm, packed with Heli-Pak stainless steel wire of132

1.8mm×1.8mm×0.2mm.133

The experiments used to validate the analytical solution were conducted in total-reflux regime134

at a pressure of approximately 0.8×105 Pa.135

The distillation column is fed with highly purified carbon monoxide in liquid state. From the136

top of the column the waste gas (12CO enriched) is withdrawn, while from the bottom of the137

column the product (13CO enriched) is taken out. A variable heating resistance (up to 150W)138

ensures the vapor stream, while the total condenser provides the reflux. The total condenser139

uses liquid nitrogen as cooling agent (N2 has a boiling point in normal conditions of 77.3K [57]).140

1For brevity, in the following, the atomic number of the oxygen is omitted, i.e. we use 13CO as a short notation

for 13C16O. Similarly, 12C16O is indicated as 12CO.

6

Figure 1: Experimental pilot-scale cryogenic distillation plant with condenser K, distillation column C, reboiler

B, vacuum jacket VJ, rough pump RP, diffusion pump DP, temperature sensors T1−T2, manometers M1−M3,

level sensor L, feed reservoir FR, buffer tank BT, and waste reservoir WR.

A multilayered vacuum jacket provides the thermal insulation. The pressure within the jacket141

was 1.07×10−2 Pa [27].142

2.3. Isotope separation PDE model143

In a previous paper [27], we have derived, using first-principles knowledge, a full nonlinear144

model for the isotope separation by cryogenic distillation. Due to the complexity of the full145

nonlinear model we have considered two alternative modeling approaches. Two simplified models146

followed, a quasi-linear model and, when the isotope concentration achieved by the separation147

plant was low, a linear model. Subsequently, we have shown that the linear model is a valid148

modeling approach for 13C isotope separation by cryogenic distillation.149

The linear model for the isotope separation process in a rectifying column is described by the150

7

following partial differential equation (PDE):151

(Hl +Hv)∂N

∂t=

(

LV

K

)

∂2N

∂z2−[

P + L(α− 1)]∂N

∂z(2)

where Hl + Hv is the total hold-up per unit volume of the raw material in the column, both152

in liquid and vapor phase, L and V stand for the liquid and vapor molar flow rates per unit153

area, K is the volumetric overall mass transfer coefficient, while P is the product flow rate given154

by2 P = L − V . The initial condition is represented by the natural abundance of 13C, while155

the boundary conditions are determined by the concentration achieved at the top of the column156

(z = 0) respectively at the bottom of the column (z = Zc) [27]. In the following, based on the157

above isotope separation PDE model, we will determine the analytical solution of the isotope158

separation process.159

3. Analytical solution of the model of the isotope separation process160

It is well known that the Laplace transform is a powerful tool for transforming initial-value161

problems involving ordinary differential equations (ODE) into algebraic equations [58, 59]. The162

Laplace transform can be also applied for transforming a PDE into another PDE with one163

independent variable less, or, if the original PDE had two independent variables, into an ODE164

[60, 61]. In this section, based on the PDE model that governs the evolution of the desired isotope165

during the isotope separation process, we will define our initial-boundary-value problem. Using166

the Laplace transform we will obtain a linear homogeneous ODE. After solving the obtained167

ODE, by applying the inverse Laplace transform we will obtain the analytical solution of the168

model of the isotope separation process.169

Fig. 2(a) shows the schematic representation of the 13C cryogenic distillation plant. The170

plant operates in extraction regime. We have denoted by F the feed flow rate, by P the product171

flow rate, and by W the waste flow rate. The feed point divides the column in two sections,172

the rectifying section and the stripping section. When the product is represented by the less173

volatile component, the rectifying section starts from the feeding point and ends at the bottom174

of the column, while the stripping section starts from the feeding point and ends at the top of175

the column3. In steady state, from the end of the rectifying section of the column, the product is176

2When the product is represented by the less volatile component.3When the product is represented by the more volatile component, the rectifying section starts from the feeding

point and ends at the top of the column, while the stripping section starts from the feeding point and ends at the

bottom of the column.

8

(a) Cryogenic distillation column. (b) Rectifying section. (c) Stripping section.

Figure 2: Schematic representation of the 13C cryogenic distillation and of the column’s rectifying and stripping

section.

taken out with a constant flow rate P and a 13C concentration equal to NP. From the end of the177

stripping section of the column, the waste is taken out with a constant flow rate W and a 13C178

concentration equal to NW. Highly purified carbon monoxide is fed the column with a constant179

flow rate F and a 13C concentration equal to the 13C natural abundance, N0. The internal liquid180

and vapor molar flow rates per unit area in the rectifying section are denoted by Lr and Vr and181

by Ls and Vs in the stripping section.182

In the rectifying section, the gradient of the height is positive by convention, while in the183

stripping section it is negative [13, 15, 23, 24]. The feeding point corresponds to height z = 0.184

The bottom of the column corresponds to the height z = Zr (Zr > 0) and the top of the column185

to the height z = Zs (Zs < 0).186

In the following we will derive the analytical solution of the isotope separation process in the187

rectifying section of the 13C cryogenic distillation column. The obtained model is applicable to188

the stripping section too; however, in that case, the height gradient will be negative.189

3.1. Defining the initial-boundary-value problem190

The linear partial differential equation that governs, with respect to both time and space, the191

evolution of 13C isotope in the liquid phase (i.e. enriched phase) in the rectifying section (see192

Fig. 2(b)) is given by [23, 27]:193

(Hl +Hv)∂N

∂t=

(

LrVrKr

)

∂2N

∂z2−[

Lr(α− 1) + P]∂N

∂z(3)

9

where Hl +Hv is the total carbon monoxide hold-up per unit volume in the rectifying section4,194

while Kr stands for the volumetric overall mass transfer coefficient in the rectifying section.195

In the case of the stripping section (see Fig. 2(c)), the partial differential equation that196

governs the 13C evolution in the liquid phase is given by :197

(Hl +Hv)∂N

∂t=

(

LsVsKs

)

∂2N

∂z2−[

Ls(α− 1)−W]∂N

∂z(4)

where Ks is the volumetric overall mass transfer coefficient in the stripping section.198

In order to solve (3) and (4) it is necessary to define the initial condition and the boundary199

conditions [37, 38]. The time-space domain of (3) and (4) is determined by the duration of the200

isotope separation process and by the length of the column.201

The initial condition in both cases is represented by the desired isotope concentration at the202

moment of time t = 0 and it is given by the natural abundance of 13C:203

N(z, t = 0) = N0 (5)

The boundary conditions are given by the desired isotope evolution at height z = 0 (feeding204

point) and at the ends of the column, i.e. the bottom of the column in the case of the rectifying205

section (z = Zr) and the top of the column in the case of the stripping section (z = Zs).206

Since during the extraction regime, the feeding is continuous and the raw material is repre-207

sented by carbon monoxide with a natural abundance of 13C, the boundary condition at height208

z = 0 is given by:209

N(z = 0, t) = N0 (6)

By integrating (3) with respect to height, we obtain:210

∫

(Hl +Hv)∂N

∂tdz + Ci =

(

LrVrKr

)

∂N

∂z−[

Lr(α− 1) + P]

N (7)

where Ci is an integration constant. The general form of (7) is referred to in [23, 24] as the211

transport equation and it relates the amount of the desired isotope carried along a column to the212

gradient of the mole fraction [24, 62]:213

T =[

Lr(α− 1) + P]

N −

(

LrVrKr

)

∂N

∂z(8)

4Since the hold-up is distributed homogeneously in the distillation column, the total carbon monoxide hold-up

per unit volume in the rectifying section is equal to the total carbon monoxide hold-up per unit volume in the

stripping section and is equal to the total carbon monoxide hold-up per unit volume in the column.

10

where T stands for the desired isotope flow rate. At the end of the isotope separation column,214

we have [23, 24, 63]:215

T = NP (9)

From (8) and (9) we obtain the boundary condition at the end of the rectifying section:216

N(z = Zr, t) =Vr

Kr(α− 1)

∂N

∂z(10)

Analogously, the boundary condition at the end of the stripping section is given by:217

N(z = Zs, t) =Vs

Ks(α− 1)

∂N

∂z(11)

Below we will use the following notations:218

ηr ≡(Hl +Hv)Kr

LrVr(12)

θr ≡Kr(α− 1)

2Vr(13)

ψr ≡P

Lr(α− 1)(14)

Using (12), (13), and (14), (3) can be written as:219

ηr∂N

∂t=∂2N

∂z2− 2θr(1 + ψr)

∂N

∂z(15)

and the boundary condition (10) becomes:220

N(z = Zr, t) =1

2θr

∂N

∂z(16)

For the stripping section, the analogous notations are:221

ηs ≡(Hl +Hv)Ks

LsVs(17)

θs ≡Ks(α− 1)

2Vs(18)

ψs ≡−W

Ls(α− 1)(19)

and (4) can be written as:222

ηs∂N

∂t=∂2N

∂z2− 2θs(1 + ψs)

∂N

∂z(20)

while the boundary condition (11) becomes:223

N(z = Zs, t) =1

2θs

∂N

∂z(21)

It can be seen that the relations for the stripping section are similar to the relations for the224

rectifying section.225

11

3.2. Laplace transform226

By applying the Laplace transform, the partial differential equation (15) turns into:227

ηr[

sG(z, s)−N0

]

=d2

dz2G(z, s)− 2θr(1 + ψr)

d

dzG(z, s) (22)

while the boundary conditions (6) and (16) become:228

G(0, s) =N0

s(23)

G(Zr, s) =1

2θr

d

dzG(Zr, s) (24)

where:229

G(z, s) = L{N(z, t)} =

∫

∞

0

N(z, t)e−stdt (25)

Since the independent variable of (22) is the height z, while s is a parameter [60, 64], in the230

following we will use the simpler notation G(z) instead of G(z, s). The relation (22) can then be231

written as:232

d2

dz2G(z)− 2θr(1 + ψr)

d

dzG(z)− ηrsG(z) = −ηrN0 (26)

It can be seen that (26) is an homogeneous linear ODE of the second order [65, 66].233

The complementary equation of (26) is:234

d2

dz2G(z)− 2θr(1 + ψr)

d

dzG(z)− ηrsG(z) = 0 (27)

So the solution of (26) has the following form:235

G(z) = Gp(z) +Gg(z) (28)

where Gp is the particular solution of (26) and Gg is the general solution of (27).236

In order to find the particular solution of (26) we will choose the method of undetermined237

coefficients. Therefore, the particular solution, which is a polynomial of the same degree as the238

non homogeneous term, will verify (26). It can be easily found that the particular solution Gp239

is given by:240

Gp =N0

s(29)

The auxiliary equation of (27) is [65]:241

r2 − 2θr(1 + ψr)r − ηrs = 0 (30)

12

with roots:242

r1 =2θr(1 + ψr) +

√

4θ2r (1 + ψr)2 + 4ηrs

2(31)

r2 =2θr(1 + ψr)−

√

4θ2r (1 + ψr)2 + 4ηrs

2(32)

Hence, the general solution of (27) is given by [66, 67]:243

Gg(z) = C1er1z + C2e

r2z (33)

where C1 and C2 are constants. The solution of (26) follows:244

G(z) = Gp(z) +Gg(z) =N0

s+ C1e

r1z + C2er2z (34)

The constants C1 and C2 can be found by replacing (34) in (23) and (24).245

For z = 0, from (34) and (23) we obtain:246

G(0) =N0

s=N0

s+ C1e

0 + C2e0 (35)

whence:247

C1 = −C2 ≡ C (36)

For z = Zr, from (34) and (24) we obtain:248

2θrG(Zr)−d

dzG(Zr) = 0 (37)

whence, referring to (34) and (36):249

G(Zr) =N0

s+ Cer1Zr − Cer2Zr (38)

The term ddzG(Zr) is given by:250

d

dzG(Zr) = r1Ce

r1Zr − r2Cer2Zr (39)

From (37), (38), and (39) we obtain:251

C(

r1er1Zr − r2e

r2Zr

)

− 2θrC(

er1Zr − er2Zr

)

= 2θrN0

s(40)

In the following we will define the term β(s) as:252

β(s) ≡√

θ2r (1 + ψr)2 + ηrs (41)

13

The roots of the auxiliary equation can then be written as:253

r1 = θr(1 + ψr) + β(s) (42)

r2 = θr(1 + ψr)− β(s) (43)

Using (42) and (43), (40) becomes:254

Ceθr(1+ψr)Zr

{

[

θr(1+ψr)+β(s)]

eβ(s)Zr−[

θr(1+ψr)−β(s)]

e−β(s)Zr−2θr

(

eβ(s)Zr−e−β(s)Zr

)}

= 2θrN0

s(44)

whence the constant C follows:255

C = 2θrN0

se−θr(1+ψr)Zr

1

β(s)(

eβ(s)Zr + e−β(s)Zr

)

− θr(1 + ψr)(

eβ(s)Zr − e−β(s)Zr

) (45)

Using Euler’s formula [43]:256

eix = cos(x) + i sin(x) (46)

whence:257

cos(x) =1

2

(

eix + e−ix)

(47)

sin(x) =1

2i

(

eix − e−ix)

(48)

we can write:258

(

eβ(s)Zr + e−β(s)Zr

)

= 2 cos(

iβ(s)Zr

)

(49)(

eβ(s)Zr − e−β(s)Zr

)

= (−2i) sin(

iβ(s)Zr

)

(50)

Referring to (49) and (50), (45) turns into:259

C = C1 = −C2 = 2θrN0

se−θr(1+ψr)Zr

1

2[

β(s) cos(

iβ(s)Zr

)

− θr(1− ψr)(−i) sin(

iβ(s)Zr

)] (51)

The solution of (26) follows:260

G(z) =N0

s+ Ceθr(1+ψr)z(−2i) sin

(

iβ(s)z)

(52)

By defining:261

γ(s) ≡ iβ(s) (53)

we can write (52) as:262

G(z, s) =N0

s+ 2θr

N0

seθr(1+ψr)(z−Zr)

sin(

γ(s)z)

γ(s) cos(

γ(s)Zr

)

− θr(1− ψr) sin(

γ(s)Zr

) (54)

14

Analogously, in the case of the striping section we obtain:263

G(z, s) =N0

s+ 2θs

N0

seθs(1+ψs)(z−Zs)

sin(

γ(s)z)

γ(s) cos(

γ(s)Zs

)

− θs(1− ψs) sin(

γ(s)Zs

) (55)

where264

γ(s) ≡ i√

θ2s (1 + ψs)2 + ηss (56)

By applying the inverse Laplace transform to (54) and (55) we obtain the analytical solution265

of the isotope separation process.266

3.3. Inverse Laplace transform and the analytical solution of the isotope separation process267

In order to find the inverse of the Laplace transform of a function, one can use the contour268

integration, the convolution theorem, tables of integral transforms, or Heaviside’s expansion269

theorem [41, 44, 45, 68–71].270

Given the complexity of (54) and (55) it is convenient to use the generalized Heaviside’s271

expansion theorem [68, 72, 73] for finding the inverse Laplace transform.272

In order to apply the Heaviside’s expansion theorem we will rewrite (54) as:273

G(z, s) =N0

s+

2θrN0eθr(1+ψr)(z−Zr)

sin(

γ(s)z)

sin(

γ(s)Zr

)

s[

γ(s) 1

tan(

γ(s)Zr

) − θr(1− ψr)] (57)

By defining274

g(s) ≡ γ(s)1

tan(

γ(s)Zr

) − θr(1− ψr) (58)

the relation (57) becomes:275

G(z, s) =

N0

[

g(s) + 2θreθr(1+ψr)(z−Zr)

sin(

γ(s)z)

sin(

γ(s)Zr

)

]

sg(s)≡q(s)

p(s)(59)

where276

q(s) ≡ N0

[

g(s) + 2θreθr(1+ψr)(z−Zr)

sin(

γ(s)z)

sin(

γ(s)Zr

)

]

(60)

and277

p(s) ≡ sg(s) (61)

The zeros of p(s) are s1 = 0 and the solutions of:278

g(s) = 0 (62)

15

or279

tan(

γ(s)Zr

)

=γ(s)

θr(1− ψr)(63)

The relation (63) is a transcendental equation and its solution will be discussed in Section280

3.4.281

In the case of the stripping section, the analogous transcendental equation is:282

tan(

γ(s)Zs

)

=γ(s)

θs(1− ψs)(64)

For finding the function that describes the evolution of the concentration of the desired283

isotope in the rectifying section, we will apply the Heaviside’s expansion theorem to (59). Thus,284

the function N will have the form [41, 60, 72]:285

N(z, t) =

n∑

k=1

akeskt (65)

where286

a1 = lims→s1

sG(z, s) (66)

while287

aj =q(sj)

p′(sj)(67)

where sj , for j = 2, 3, . . . , n, are the solutions of (63).288

Theorem 1. Final value theorem [59]. If F is the Laplace transform of the function f , then:289

lims→0

sF (s) = limt→∞

f(t) (68)

It can be seen that the value of the coefficient a1 is the final value of the function N :290

a1 = lims→0

sG(z, s) = limt→∞

N(z, t) (69)

By solving (66), using (68), we obtain the stationary value of the concentration of the desired291

isotope at an arbitrary height z in the rectifying section:292

limt→∞

N(z, t) = N0e2θr(1+ψr)z + ψre

2θr(1+ψr)Zr

1 + ψre2θr(1+ψr)Zr

(70)

Analogously, the stationary value of the concentration of the desired isotope at an arbitrary293

height z in the stripping section is given by:294

limt→∞

N(z, t) = N0e2θs(1+ψs)z + ψse

2θs(1+ψs)Zs

1 + ψse2θs(1+ψs)Zs

(71)

16

Since sj are the solution of g(s) = 0, (60) yields:295

q(sj) = N0

[

2θreθr(1+ψr)(z−Zr)

sin(

γ(sj)z)

sin(

γ(sj)Zr

)

]

(72)

and296

p′(sj) = sjg′(sj) (73)

By derivation with respect to s, (58) becomes:297

g′(sj) =ηrZr

2−

ηr

2(

γ(sj))2 θr(1− ψr)

[

1− Zrθr(1− ψr)]

(74)

From (72) and (74) we obtain the value of the coefficient aj :298

aj =

N0

[

2θreθr(1+ψr)(z−Zr)

sin(

γ(sj)z)

sin(

γ(sj)Zr

)

]

sj

{

ηrZr

2 − ηr2(γ(sj))2

θr(1− ψr)[

1− Zrθr(1− ψr)]

} (75)

The analytical solution of the isotope separation process, in the rectifying section, follows:299

N(z, t) = N0e2θr(1+ψr)z + ψre

2θr(1+ψr)Zr

1 + ψre2θr(1+ψr)Zr

+

n∑

j=2

N0

[

2θreθr(1+ψr)(z−Zr)

sin(

γ(sj)z)

sin(

γ(sj)Zr

)

]

sj

{

ηrZr

2 − ηr2(γ(sj))2

θr(1− ψr)[

1− Zrθr(1− ψr)]

}esjt

(76)

where θr, ψr, and γ(sj) are the parameters of the isotope separation process in the rectifying300

section (see (12), (13), (14), and (53)), while sj , for j = 2, 3, . . . , n, are the solutions of the301

transcendental equation (63).302

Analogously, in the case of the stripping section, the analytical solution is given by the similar303

relation:304

N(z, t) = N0e2θs(1+ψs)z + ψse

2θs(1+ψs)Zs

1 + ψse2θs(1+ψs)Zs

+n∑

j=2

N0

[

2θseθs(1+ψs)(z−Zs)

sin(

γ(sj)z)

sin(

γ(sj)Zs

)

]

sj

{

ηsZs

2 − ηs2(γ(sj))2

θs(1− ψs)[

1− Zsθs(1− ψs)]

}esjt

(77)

In this case, θs, ψs, and γ(sj) are the parameters of the isotope separation process in the305

stripping section (see (17), (18), (19), and (53)), while sj , for j = 2, 3, . . . , n, are the solutions of306

the transcendental equation (64).307

Using (76) and (77) one can determine the concentration of the desired isotope at any height308

z and at any moment of time t, with respect to plant parameters like the hold-up, liquid and309

vapor molar flow rate, separation factor, and the height of the distillation column.310

17

Figure 3: Graphical representation of the functions y1(

γ(s)Zr

)

, y1(

γ(s)Zs

)

and y2(

γ(s)Zr

)

, y2(

γ(s)Zs

)

.

3.4. Solving the transcendental equations (63) and (64)311

In our case, (63) can be written as:312

tan(

γ(s)Zr

)

=

(

γ(s)Zr

)

θrZr(1− ψr)(78)

Both sides of (78) can be expressed as two functions of the same argument:313

y1(

γ(s)Zr

)

= tan(

γ(s)Zr

)

(79)

y2(

γ(s)Zr

)

=

(

γ(s)Zr

)

Zrθr(1− ψr)(80)

Fig. 3 shows the graphical representation of the functions y1(

γ(s)Zr

)

, y1(

γ(s)Zs

)

and314

y2(

γ(s)Zr

)

, y2(

γ(s)Zs

)

. Since the height gradient is positive in the rectifying section and negative315

in the stripping section, y′2(

γ(s)Zr

)

> 0, while y′2(

γ(s)Zs

)

< 0.316

The solutions of (78) are given by the points for which the following is true:317

y1(

γ(sj)Zr

)

= y2(

γ(sj)Zr

)

(81)

18

or, in other words, in the intersection points of the two functions.318

By knowing the value of(

γ(sj)Zr

)

, where j = 2, 3, . . . , n, one can find the solutions of (62),319

and the analytic solution of the isotope separation process in the rectifying section becomes320

completely determined. Knowing that (see (53)):321

(

γ(sj)Zr

)

= i

(

√

θ2r (1 + ψr)2 + ηrsj

)

Zr (82)

we obtain the solutions of (62):322

sj = −

[

(

γ(sj)Zr

)2+ Zr

2θ2r (1 + ψr)2

ηrZr2

]

(83)

Since ηr is a positive quantity (see (12)) one can see from (83) that sj < 0. Thus, the terms323

in the infinite series (see (76)) are transient and the series converges.324

Analogously, in the case of the stripping section we obtain:325

sj = −

[

(

γ(sj)Zs

)2+ Zs

2θ2s (1 + ψs)2

ηsZs2

]

(84)

For determining the intersection points(

γ(sj)Zr

)

and(

γ(sj)Zs

)

as plotted in Fig. 3 we have326

used the MATLAB function fzero. The function fzero find the roots of a continuous function of327

one variable around an arbitrary point [74, 75].328

4. Validation of the analytical solution of the isotope separation process329

Usually, the only points where the isotopic concentration is measurable are at the bottom and330

at the top of the column. In order to validate the analytical solution, we will use (76) and (77)331

written for the boundary conditions z = Zr (the end of the rectifying section, i.e. the bottom of332

the column) and z = Zs (the end of the stripping section, i.e. the top of the column).333

In the following we will refer to three isotope separation experiments carried using the exper-334

imental pilot-scale plant developed at the National Institute for Research and Development of335

Isotopic and Molecular Technologies in Cluj-Napoca, Romania. The three experiments were con-336

ducted in total-reflux regime for different electrical powers that supplied the heating resistance,337

i.e. 24W, 27W, and 29W.338

The analytical solution (76) written for z = Zr is given by:339

N(z = Zr, t) = N0(1 + ψr)e

2θr(1+ψr)Zr

1 + ψre2θr(1+ψr)Zr

+

n∑

j=2

N02θr

sj

{

ηrZr

2 − ηr2(γ(sj))2

θr(1− ψr)[

1− Zrθr(1− ψr)]

}esjt

(85)

19

Since the experiments were conducted in total-reflux regime, no withdrawal was performed340

and therefore:341

F = P =W = 0 (86)

whence (see (14)) ψr = 0. Thus, (85) becomes:342

N(z = Zr, t) = N0e2θrZr +

n∑

j=2

N02θr

sj

[

ηrZr

2 − ηrθr2(γ(sj))2

(1− Zrθr)]esjt (87)

Analogously, the analytical solution written for the boundary condition z = Zs and the343

total-reflux condition ψs = 0 is given by:344

N(z = Zs, t) = N0e2θsZs +

n∑

j=2

N02θs

sj

[

ηsZs

2 − ηsθs2(γ(sj))2

(1− Zsθs)]esjt (88)

The relations (87) and (88) describe the evolution of the mole fraction of the desired isotopic345

compound at the bottom and at the top of the column with respect to process parameters like346

hold-up, internal molar flow rates, height of the column, and the separation factor.347

4.1. Parameters of the analytical solution of the isotope separation process348

Since the plant’s configuration consists of one distillation column with a single reboiler, the349

vapor molar flow rate in the rectifying section is equal to the vapor molar flow rate in the stripping350

section (Vr = Vs). The vapor internal stream is determined by the electrical power that supplied351

the heating resistance and the heat transfer through the multilayered vacuum jacket [76–78].352

The liquid molar flow rate is determined by the total-reflux condition. The total hold-up in the353

column Hl + Hv is determined by measuring the quantity of raw material fed to the column354

during the operations prior to the beginning of each experiment.355

Naturally, during the total-reflux experiment, the raw material will have, at a certain height356

z, in steady state, the natural isotopic abundance (N0). When the product is represented by357

the less volatile component, under the height corresponding to the natural isotopic abundance358

(toward to the bottom of the column) the concentration of the desired isotope will be higher than359

its natural abundance, while above this height, the isotope concentration will be lower than its360

natural abundance. Therefore, the total-reflux regime can be analyzed as a particular case of the361

extraction regime, when no withdrawal is performed and thus, the feed flow rate F , the product362

flow rate P , and the waste flow rate W are zero, and the feed point will correspond to the height363

for which the raw material isotopic concentration corresponds to the natural abundance (see Fig.364

2(a), Fig. 2(b), and Fig. 2(c)).365

20

The rate of transfer of the heavy isotopic compound (the product) to the liquid phase across366

the interface per unit volume, in the rectifying section, is given by [23, 27]:367

τr = −Kr

[

(N − n)− n(α− 1)(1−N)]

(89)

where the separation factor α is:368

α =N

1−Nn

1−n

(90)

Analogously, the rate of transfer of the light isotopic compound to the vapor phase in the369

stripping section is given by:370

τ ′s = −Ks

[

(n′ −N ′)−N ′(α′ − 1)(1− n′)]

(91)

where n′ and N ′ correspond to the mole fraction of the light isotopic compound in the vapor371

phase, while α′ is:372

α′ =n′

1−n′

N ′

1−N ′

> 1 (92)

Since during the total-reflux regime no products are withdrawn and the feed flow rate is zero,373

the rate of transfer of the heavy isotopic compound to the liquid phase in the stripping section374

will be equal but in opposite sign to the rate of transfer of the heavy isotopic compound to the375

liquid phase in the rectifying section:376

τs = τ ′s = −τr (93)

In our case, the raw material corresponds to a binary mixture, and thus [27, 32]:377

n′ = 1− n (94)

N ′ = 1−N (95)

α′ = α (96)

The description of τs follows:378

τs = −Ks

[

(N − n)− n(α− 1)(1−N)]

(97)

From (89), (93), and (97) we obtain:379

Kr = Ks (98)

Previously we have determined the stationary value of the concentration of the desired isotope380

at an arbitrary height z in the separation column (see (70) and (71)). Usually, the only points381

21

were the isotopic concentration is measurable are at the top of the column and at the bottom of382

the column. By referring to (70) and (71) and knowing that the experiments were conducted in383

total-reflux regime and therefore ψr = ψs = 0 (see (14) and (19)) we obtain the stationary value384

of the concentration at the end of the rectifying section z = Zr (bottom of the column) and at385

the end of the stripping section z = Zs (top of the column):386

limt→∞

N(Zr, t) = N0e2θrZr (99)

limt→∞

N(Zs, t) = N0e2θsZs (100)

Since the stationary values of the desired isotope concentration are determined experimentally,387

by referring to (13) and (18), the description of Kr and Ks follows:388

Kr =Vr

(α− 1)Zrln

(

limt→∞

N(Zr, t)

N0

)

(101)

Ks =Vs

(α− 1)Zsln

(

limt→∞

N(Zs, t)

N0

)

(102)

From (98), (101), and (102), and knowing that Zc = Zr − Zs is possible to determine the389

height of the rectifying and stripping section of a distillation column that operates in total-reflux390

regime:391

Zr = Zc

ln

(

limt→∞

N(Zr,t)

N0

)

ln

(

limt→∞

N(Zr,t)

limt→∞

N(Zs,t)

) (103)

Zs = Zc

ln

(

limt→∞

N(Zs,t)

N0

)

ln

(

limt→∞

N(Zr,t)

limt→∞

N(Zs,t)

) (104)

Since the heights of the rectifying and stripping section (Zr and Zs) depend on measurable392

parameters like the total height of the column Zc, the stationary values of the concentration at393

the ends of the column, and on the natural abundance of 13C, one can determine the constant394

concentration point Z0 as being at height Zs from the top of the column or Zr from the bottom395

of the column (see Figure 2(a)).396

Knowing the vapor and liquid molar flow rates, the total hold-up in the column (Hl +Hv),397

the heights of the rectifying and stripping section (Zr and Zs), and the separation factor (α), it398

is possible to determine the parameters of the analytical solution and, thus, to obtain the isotope399

distribution with respect to both time and height by computing (76) and (77).400

22

Table 2: Parameters of the analytical solution and the heights of the rectifying and stripping section in the case

of all three isotope separation experiments.

Pelectrical ηr = ηs θr = θs ψr = ψs Zr Zs

(W)(

s

m2

) (

1

m

)

(-) (m) (m)

24 12198.062 0.168 0 2.390 -4.610

27 9664.195 0.143 0 2.956 -4.044

29 9798.104 0.152 0 2.884 -4.116

Table 3: The values of(

γ(sj)Zr

)

and(

γ(sj)Zs

)

when j = 2, 3, 4, 5, and 6 in the case of all three isotope

separation experiments.

Pelectrical

(

γ(s2)Zr

) (

γ(s3)Zr

) (

γ(s4)Zr

) (

γ(s5)Zr

) (

γ(s6)Zr

)

(W) (−) (−) (−) (−) (−)

24 1.262 4.625 7.802 10.958 14.108

27 1.242 4.621 7.799 10.956 14.107

29 1.226 4.617 7.797 10.955 14.105

Pelectrical

(

γ(s2)Zs

) (

γ(s3)Zs

) (

γ(s4)Zs

) (

γ(s5)Zs

) (

γ(s6)Zs

)

(W) (−) (−) (−) (−) (−)

24 1.949 4.870 7.951 11.065 14.191

27 1.870 4.831 7.926 11.047 14.177

29 1.891 4.841 7.933 11.052 14.181

Table 2 summarizes the parameters of the analytical solution, as well as the heights of the401

rectifying section and stripping section in the case of the three isotope separation experiments.402

The first five values5 of (γ(sj)Zr) and (γ(sj)Zs) are listed in Table 3.403

4.2. Assessment of the analytical solution of the isotope separation process404

Fig. 4(a), 4(b), and 4(c) show the 13CO mole fraction evolution at both ends of the distillation405

column in the case of the three isotope separation experiments. The results were obtained by406

computing the analytical solution described by (87) and (88). In addition, the measured values407

are also plotted.408

5For brevity, we remind that the first solution of the transcendental equations is indicated by the index j = 2.

23

0 15 30 45 60 75 90 105 120 1350

0.235

1.11

2.482

3

time (h)

13C

conce

ntr

atio

n(%

)

bottom isotope concentration evolution (analytic model)

bottom sample composition

top isotope concentration evolution (analytic model)

top sample composition

(a) 24W case.

0 12 24 36 48 60 72 820

0.349

1.11

2.586

3

time (h)

13C

conce

ntr

atio

n(%

)

bottom isotope concentration evolution (analytic model)

bottom sample composition

top isotope concentration evolution (analytic model)

top sample composition

(b) 27W case.

0 12 24 36 48 60 72 840

0.316

1.11

2.6773

time (h)

13C

conce

ntr

atio

n(%

)

bottom isotope concentration evolution (analytic model)

bottom sample composition

top isotope concentration evolution (analytic model)

top sample composition

(c) 29W case.

Figure 4: 13C concentration evolution according to the analytical solution and 13C concentrations achieved by

the pilot-scale experimental plant.

Fig. 5(a), 5(b), and 5(c) show the relative error of the analytical solution at both ends of the409

separation column. Explicitly, in all three cases, the relative error was calculated using:410

ζ(z = Zr, t) =

∣

∣

∣

∣

∣

Nanalytic(z = Zr, t)−Nexperimental(z = Zr, t)

Nanalytic(z = Zr, t)

∣

∣

∣

∣

∣

(105)

24

0 15 30 45 60 75 90 105 120 1350

20.738

23.09225

time (h)

rela

tive

erro

r(%

)

relative error at the end of the rectifying section (z = Zr)

relative error at the end of the stripping section (z = Zs)

(a) 24W case.

0 12 24 36 48 60 72 820

3.723

11.155

25

time (h)

rela

tive

erro

r(%

)

relative error at the end of the rectifying section (z = Zr)

relative error at the end of the stripping section (z = Zs)

(b) 27W case.

0 12 24 36 48 60 72 840

7.046

22.289

25

time (h)

rela

tive

erro

r(%

)

relative error at the end of the rectifying section (z = Zr)

relative error at the end of the stripping section (z = Zs)

(c) 29W case.

Figure 5: Relative errors between simulated and measured data at the end of the rectifying section (bottom of

the column) and at the end of the stripping section (top of the column).

for the rectifying section. For the stripping section the relative error was calculated similarly.411

Fig. 6(a), 6(b), 6(c), 6(d), 6(e), and 6(f) show the 13CO mole fraction distribution in the412

column with respect to both height and time. The isotope distribution was obtained by com-413

puting the analytical solution described by (76) and (77). The simulation results were obtained414

25

01

23

45

67

015

3045

6075

90105

120135

00.235

1.11

2.482

3

height (m)time (h)

13C

conce

ntr

ation

(%)

0.235

1.11

2.482

(a) 13C isotope concentration distribution. The electrical

power applied to the heater was 24W.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

height (m)

tim

e(h

)

0 1 2 3 4 5 6 70

15

30

45

60

75

90

105

120

135

0.235

1.11

2.482

(b) 13C isotope concentration distribution (contour view).

The electrical power applied to the heater was 24W.

01

23

45

67

012

2436

4860

7282

00.349

2.586

3

height (m)time (h)

13C

conce

ntr

atio

n(%

)

0.349

1.11

2.586

(c) 13C isotope concentration distribution. The electrical

power applied to the heater was 27W.

height (m)

tim

e(h

)

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22.42.4

0 1 2 3 4 5 6 70

12

24

36

48

60

72

82

0.349

1.11

2.586

(d) 13C isotope concentration distribution (contour view).

The electrical power applied to the heater was 27W.

01

23

45

67

012

2436

4860

7284

00.316

1.11

2.6773

height (m)time (h)

13C

conce

ntr

atio

n(%

)

0.316

1.11

2.677

(e) 13C isotope concentration distribution. The electrical

power applied to the heater was 29W.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22.4

height (m)

tim

e(h

)

0 1 2 3 4 5 6 70

12

24

36

48

60

72

84

0.316

1.11

2.677

(f) 13C isotope concentration distribution (contour view).

The electrical power applied to the heater was 29W.

Figure 6: 13C isotope concentration distribution with respect to time and height.

for a 70 discretization divisions applied to the space domain (i.e. a height step of 0.1m) and a415

time step of 1800 s emphasizing that the isotopic concentration can be simulated at any moment416

of time and at any height. The simulation time was approximately 1.5 s in the case of all three417

experiments.418

In Table 4 we have listed the values of the maximum and average relative error at the ends419

of the separation column, in the case of all three experiments.420

26

Table 4: The values of the maximum and average relative error between simulated and measured data at the ends

of the separation column.

Pelectrical max(

ζ(z = Zr, t))

max(

ζ(z = Zs, t))

ζ(z = Zr, t) ζ(z = Zs, t)

(W) (%) (%) (%) (%)

24 20.738 23.092 5.880 6.696

27 3.723 11.155 1.144 4.516

29 7.046 22.289 2.225 8.318

When the electrical power applied to the heater was 24W, the average relative error at the421

bottom of the column was 5.8%, while at the top of the column it was 6.6%. When the electrical422

power was 27W, the average relative error was 1.1% at the bottom of the column and 4.5% at423

the top of the column. In the case of 29W, the average relative error was 2.2% at the bottom424

and 8.3% at the top of the column.425

The volumetric overall mass transfer coefficient is determined by plant parameters such as426

the internal vapor molar flow rate V , the separation factor α, the heights of the rectifying and427

stripping sections of the distillation column (Zr and Zs), and to the steady-state values of the428

concentration achieved at the ends of the separation column. Using the results of Section 4, one429

can use interpolation to determine the volumetric overall mass transfer coefficient with respect430

to the above parameters, and thus, expensive experiments could be avoided.431

5. Conclusions432

Isotope separation processes have a coupled time-space nature and their inputs, outputs, and433

parameters can vary both in time and space. Modeling of such complex processes requires time434

and knowledge from a variety of fields. However, finding an appropriate analytic model eases435

the linking to the physics of the process and brings considerable advantages in understanding436

the subprocesses and the associated interactions. In this paper we have presented a modeling437

approach in order to find the analytic expression of a 13C isotope separation process by cryogenic438

distillation of carbon monoxide. Firstly, after a short review of isotope separation processes, based439

on the partial differential equation that governs the isotope separation process, we have defined440

the initial-boundary-value problem. Using the Laplace transform, we have transformed the PDE441

into a linear homogeneous ODE. Next, we have determined the solution of the ODE and, by using442

Heaviside’s expansion theorem, we have found the inverse Laplace transform. The analytic model443

27

of the isotope separation process followed. The only points where the isotopic concentration is444

measurable in practice are at the ends of the distillation column. In order to validate the analytic445

model, we have simulated the evolution of the desired isotope in total-reflux regime, at both the446

bottom and the top of the column, and compared it with experimental data gathered from an447

experimental 13C isotope separation plant. The three experiments were conducted in total-reflux448

regime for different electrical powers that supplied the heating resistance, i.e. 24W, 27W, and449

29W. When the electrical power was 24W, the average relative error at the bottom of the column450

was 5.8%, while at the top of the column it was 6.6%. In the case of 27W, the average relative451

errors were 1.1% at the bottom of the column and 4.5% at the top of the column. When the452

electrical power was 29W, the average relative error at the bottom of the column was 2.2% and453

8.3% at the top of the column, showing that the analytic model is a valid modeling approach.454

Future studies will involve withdrawal-regime isotope separation experiments, in order to455

further analyze the analytic model and the volumetric overall mass transfer coefficient. Also, the456

effects of the packing over the hold-up and response time, internal flow rates, and production457

will be treated in order to optimize the isotope separation process. Using the obtained analytic458

model, the effects of the hydrodynamical variables fluctuations over the separation process will459

be studied from the control point of view.460

Acknowledgments461

This paper was supported by the project ”Doctoral studies in engineering sciences for devel-462

oping the knowledge based society-SIDOC” contract no. POSDRU/88/1.5/S/60078, project463

co-funded from European Social Fund through Sectorial Operational Program Human Re-464

sources 2007-2013.465

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